A combinatorial integration on the Cantor dust

Cyclic cohomology for algebras [2] is a fundamental tool to study noncommutative geometry. One of its application is the study of fractal sets to which powerful tools such as de Rham homology (on smooth manifolds) cannot be applied. $K$-homology, one of other homotopy invariant theories, of fractal sets is also studied extensively. For some class of fractal sets such as the Cantor set and the Cantor dust, the $K^{0}$ homology groups are isomorphic to $\displaystyle\prod^{\infty}\mathbb{Z}$, which is known as the Baer-Specker group [1]. One feature of the Baer-Specker group is that the group does not admit basis. This feature makes the study of $K^{0}$ groups for such fractal sets somewhat intractable because local topological features of the spaces cannot induce local algebraic structures. Cyclic cohomology on the one hand is expected to be favorable in these cases because it can be characterized as a “linearization” of $K$-homology through the Chern character.

There is a study [4] by H. Moriyoshi and T. Natsume which presented a variant of the Riemann-Stieltjes integration on the middle third Cantor set. Let $CS=\displaystyle\bigcap_{n=0}^{\infty}I_{n}$ be the middle third Cantor set, where $I_{0}=[0,1]$, $I_{1}=\left[0,\dfrac{1}{3}\right]$, $I_{2}=\left[0,\dfrac{1}{3^{2}}\right]\cup\left[\dfrac{2}{3^{2}},\dfrac{3}{3^{2}}\right]\cup\left[\dfrac{6}{3^{2}},\dfrac{7}{3^{2}}\right]\cup\left[\dfrac{8}{3^{2}},1\right]$, …. We also denote by $(H_{n},F_{n})$ the Fredholm module on $I_{n}$ which is defined by the direct sum of Connes’ Fredholm module on intervals. In [4], a new class of algebra is introduced: an algebra $\mathcal{P}=c^{\ast}BV(S^{1})$ defined by the pull-back of the bounded variation class on a unit circle $BV(S^{1})$ with the canonical Cantor function $c$. They defined a functional $\psi(f,g)=\displaystyle\lim_{n\to\infty}\psi_{n}(f,g)=\lim_{n\to\infty}\mathrm{Tr}(\epsilon f[F_{n},g]F_{n})$ on $\mathcal{P}$. The existence of the limit can be proved by showing that the cocycle is reduced to the Riemann-Stieltjes integration. A key ingredient for the proof is the Cantor function (and its intermediate functions that converge into the Cantor function); the function is used to map the iterated function system of Cantor sets onto $2$-fold subdivisions of the unit interval on which the Riemann-Stieltjes integration is defined. Because the Cantor function remains surjective onto the unit interval when the function is restricted to the Cantor set, the iterated function system for the Cantor set may be seen as a variant of subdivisions on $I$.

In the present paper, by following the idea mentioned above, we construct a new cyclic $2$-cocycle on the Cantor dust $CD$ and show that the cocycle is not trivial. In order to construct the cocycle, we define a sequence of functionals $\phi_{n}$ (see Definition 3.2), that is a generalization of a functional $\psi_{n}$ on $CS$, by using a combinatorial Fredholm module on squares defined by the authors [3] and use the product of the Cantor function. The function induces the map $\boldsymbol{c}$ between the Cantor dust and $2$-torus $\mathbb{T}^{2}$:

We can take the value $\phi_{n}(f,g,h)$ for Lipschitz functions $f,g,h\in C^{\mathrm{Lip}}(CD)$ by the definition of $\phi_{n}$. However, we have $\displaystyle\lim_{n\to\infty}\phi_{n}(f,g,h)=0$ for any $f,g,h\in C^{\mathrm{Lip}}(CD)$ (see Proposition 3.3). Thus the algebra $\mbox{$\boldsymbol{c}$}^{*}C^{\infty}(\mathbb{T}^{2})$ is far different from $C^{\mathrm{Lip}}(CD)$.

All the main results to be shown also hold for the Sierpinski carpet. $\mathbb{T}^{2}$ can be also replaced with a $2$-dimensional sphere. In order to generalize our main theorem to general dimension, we need a general method to prove the existence of a cyclic cocycle. We will leave the study for future work.

In this section, we review a combinatorial Fredholm module on squares constructed by the authors [3]. Let $\gamma\subset\mathbb{R}^{2}$ be a square of dimension $2$ and $V=\{v_{0},\ v_{1},\ v_{2},\ v_{3}\}$ the set of vertices of $\gamma$; see the following figure for the numbering of the vertices.

Set $V_{0}=\{v_{0},\ v_{2}\}$ and $V_{1}=\{v_{1},\ v_{3}\}$, so we have $V=V_{0}\cup V_{1}$. Set

and $\mathcal{H}=\mathcal{H}^{+}\oplus\mathcal{H}^{-}$. The vector space $\mathcal{H}(\cong\mathbb{C}^{4})$ is a Hilbert space of dimension $4$ with an inner product

We assume that $\mathcal{H}$ is $\mathbb{Z}_{2}$-graded with the grading $\epsilon=\pm 1$ on $\mathcal{H}^{\pm}$, respectively. The $C^{\ast}$-algebra $C(V)$ of continuous functions on $V$ acts on $\mathcal{H}$ by multiplication:

Set $U=\dfrac{1}{\sqrt{2}}\begin{bmatrix}1&-1\\ 1&1\end{bmatrix}$ and $F=\begin{bmatrix}&U^{\ast}\\ U&\end{bmatrix}=\dfrac{1}{\sqrt{2}}\begin{bmatrix}&&1&1\\ &&-1&1\\ 1&-1&&\\ 1&1&&\end{bmatrix}$. Then $F$ is a bounded operator on $\mathcal{H}$ and we have $F\epsilon+\epsilon F=O$. So $(\mathcal{H},F)$ is the Fredholm module on $C(V)$.

Set $I=[0,1]\times[0,1]$ and let $f_{s}:I\to I$ ($s=1,\dots,N$) be similitudes. Denote by

the similarity ratio of $f_{s}$. An iterated function system (IFS) $(I,S=\{1,\dots,N\},\{f_{s}\}_{s\in S})$ defines the unique non-empty compact set $K=K(\gamma_{n},S=\{1,\dots,N\},\{f_{s}\}_{s\in S})$ called the self-similar set such that $K=\bigcup_{s=1}^{N}f_{s}(K)$. Denote by $\dim_{S}(K)$ the similarity dimension of $K$, that is, the number $s$ that satisfies

If an IFS $(I,S,\{f_{s}\}_{s\in S})$ satisfies the open set condition, we have $\dim_{H}(K)=\dim_{S}(K)$, where we denote by $\dim_{H}(K)$ the Hausdorff dimension of $K$.

Set $f_{\mbox{$\boldsymbol{s}$}}=f_{s_{1}}\circ\dots\circ f_{s_{j}}$ for $\mbox{$\boldsymbol{s}$}=(s_{1},\dots,s_{j})\in S^{\infty}=\bigcup_{j=0}^{\infty}S^{\times j}$ and $f_{\emptyset}=\mathrm{id}$. For the simplicity, we will denote by $i$ the vertex $f_{\mbox{$\boldsymbol{s}$}}(v_{i})$ of a square $f_{\mbox{$\boldsymbol{s}$}}(I)$. We also denote by $V_{\mbox{$\boldsymbol{s}$}}$ the vertices of a square $f_{\mbox{$\boldsymbol{s}$}}(I)$. Denote by $e_{\mbox{$\boldsymbol{s}$}}$ the length of edge of $f_{\mbox{$\boldsymbol{s}$}}(I)$, which equals $\prod_{k=1}^{j}r_{s_{k}}$. As introduced above, we set the Hilbert space $\mathcal{H}_{\mbox{$\boldsymbol{s}$}}=\ell^{2}(V_{\mbox{$\boldsymbol{s}$}})$ on $f_{\mbox{$\boldsymbol{s}$}}(I)$ of the length $e_{\mbox{$\boldsymbol{s}$}}$, which splits the positive part $\mathcal{H}_{\mbox{$\boldsymbol{s}$}}^{+}$ and the negative part $\mathcal{H}_{\mbox{$\boldsymbol{s}$}}^{-}$. Taking direct sum on squares, we set as follows:

The $C^{\ast}$-algebra $C(K)$ acts on $\mathcal{H}_{n}$ by multiplication:

We introduce an IFS of the Cantor dust and $2$-dimensional analogue of the Cantor function defined on $1$-dimensional interval. The IFS of the Cantor dust is our main interest in the paper. The analogue of the Cantor function plays a crucial role in construction of cyclic cocycle in Section 3. Let $I=[0,1]\times[0,1]$. The IFS of the Cantor dust is defined as a set of functions $\{f_{s}:I\rightarrow\mathbb{R}^{2}\}_{s=1,2,3,4}$ described as follows:

This IFS induces a unique non-empty compact set $CD$ in $I$ such that $\displaystyle CD=\bigcup_{i=1}^{4}f_{s}(CD)$ holds; $CD$ is called the Cantor dust.

Let $c_{0}=x:[0,1]\rightarrow\mathbb{R}$. We define in an inductive manner a sequence of $\mathbb{R}$-valued continuous functions defined on the unit interval $\{c_{n}\}_{n\in\mathbb{N}}$ as follows:

The limit of $\{c_{n}\}_{n\in\mathbb{N}}$ exists and is called the Cantor function. We then generalize the construction to $2$-dimensional case by taking the pair of the two identical sequences:

The limit of the sequence $\{\mbox{$\boldsymbol{c}$}_{n}\}$ also exists and equals the product of the Cantor functions. We call the limit $\displaystyle\lim_{n\to\infty}\mbox{$\boldsymbol{c}$}_{n}$ $2$-dimensional Cantor dust function. We note that the construction of the $2$-dimensional Cantor dust function can be generalized to higher dimensional case.